Time allowed: 20 minutes.
1. (3 marks)
It is known that the vector
⎛
⎜
⎜
⎝
2
0
1
k
⎞
⎟
⎟
⎠
is in span
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎛
⎜
⎜
⎝
1
2
3
4
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
3
6
9
12
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
3
3
3
0
⎞
⎟
⎟
⎠
,
⎛
⎜
⎜
⎝
1
−
1
−
4
−
8
⎞
⎟
⎟
⎠
⎫
⎪
⎪
⎬
⎪
⎪
⎭
.
(a) Determine the value of
k
.
(b) Hence or otherwise, for the value of
k
determined in part (a) above, find values of
λ
1
, λ
2
, λ
3
, λ
4
such that:
⎛
⎜
⎜
⎝
6
0
3
3
k
⎞
⎟
⎟
⎠
=
λ
1
⎛
⎜
⎜
⎝
1
2
3
4
⎞
⎟
⎟
⎠
+
λ
2
⎛
⎜
⎜
⎝
3
6
9
12
⎞
⎟
⎟
⎠
+
λ
3
⎛
⎜
⎜
⎝
3
3
3
0
⎞
⎟
⎟
⎠
+
λ
4
⎛
⎜
⎜
⎝
1
−
1
−
4
−
8
⎞
⎟
⎟
⎠
.
2. (3 marks)
We would like to fit the quadratic curve
y
=
β
1
+
β
2
x
+
β
3
x
2
to a set of points (
x
1
, y
1
)
,
(
x
2
, y
2
)
, . . . ,
(
x
n
, y
n
)
by the method of least squares.
(a) Write down the normal equations in matrix form.
(b) For this dataset,
n
= 12
,
∑
x
i
= 22
,
∑
x
2
i
= 11
,
∑
x
3
i
= 6
,
∑
x
4
i
= 1
,
∑
x
5
i
= 0
.
5
,
∑
y
i
= 45
,
∑
x
i
y
i
= 38
,
∑
x
2
i
y
i
= 22
,
∑
x
i
y
2
i
= 35
,
∑
x
2
i
y
2
i
= 58.
Show that
β
1
= 1
, β
2
= 2 and
β
3
=
−
1.
3. (4 marks)
Suppose that
˜
v
1
=
⎛
⎜
⎜
⎝
1
0
0
1
⎞
⎟
⎟
⎠
,
˜
v
2
=
⎛
⎜
⎜
⎝
1
0
1
0
⎞
⎟
⎟
⎠
,
˜
v
3
=
⎛
⎜
⎜
⎝
1
1
1
1
⎞
⎟
⎟
⎠
and
π
= span(
˜
v
1
,
˜
v
2
).
(a) Find an orthogonal basis for
π
.
(b) Hence, or otherwise, find proj
π
˜
v
3
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Three '11
 Linear Algebra, Yi, orthogonal basis

Click to edit the document details